Problem: Mark charged a battery. Each minute, the percentage of the battery's capacity that was charged increased by $2.5$ percent. After $17$ minutes of charging, the battery was $61.5$ percent full. How full was the battery when the charging began?
The percentage of the battery's capacity that was charged increased by $2.5$ percent each minute, so it increased by $2.5T$ percent in $T$ minutes. The percentage of the battery's capacity that was charged at any given time is found by taking the percentage the battery was already charged when the charging began and adding to it the percentage that was additionally charged since then. We can express this with the equation $C=A+2.5T$, where: $C$ represents the percentage of the battery's capacity that is charged at a given time $A$ represents the percentage of the battery's capacity that was charged when the charging began $T$ represents the time (in minutes) We want to find $A$, so let's first solve the equation for $A$ : $ \begin{aligned}C&=A+2.5T\\ A&=C-2.5T\end{aligned}$ Now, we know that after $17$ minutes of charging $(T={17})$, the battery was $61.5$ percent full $(C={61.5})$. Let's plug these values into the equation to find the value of $A$. $ A={61.5}-2.5\cdot{17}=19$ Therefore, when the charging began, the battery was $19$ percent full. To find how long it took the battery to be fully charged, we can plug $C=100$ into the equation and solve for $T$. $ \begin{aligned}19&=100-2.5T\\ 2.5T&=81\\ T&=32.4\end{aligned}$ When the charging began, the battery was $19$ percent full. It took the battery $32.4$ minutes to be fully charged.